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Lagrangian relaxation hybrid with evolutionary algorithm for short-term generation scheduling
Electrical Power and Energy Systems 64 (2015) 356–364
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Lagrangian relaxation hybrid with evolutionary algorithm for short-term generation scheduling Thillainathan Logenthiran ⇑, Wai Lok Woo, Van Tung Phan School of Electrical and Electronic Engineering, Newcastle University, Singapore Campus, 180 Ang Mo Kio Avenue 8, Block P, Level 2, Unit 220 (Room P.220B), Singapore 569830, Singapore
a r t i c l e
i n f o
Article history: Received 14 November 2013 Received in revised form 30 June 2014 Accepted 17 July 2014
Keywords: Short-term generation scheduling Profit-based unit commitment Cost-based unit commitment Lagrangian relaxation Evolutionary algorithm Economic dispatch
a b s t r a c t Short-term generation scheduling is an important function in daily operational planning of power systems. It is defined as optimal scheduling of power generators over a scheduling period while respecting various generator constraints and system constraints. Objective of the problem includes costs associated with energy production, start-up cost and shut-down cost along with profits. The resulting problem is a large scale nonlinear mixed-integer optimization problem for which there is no exact solution technique available. The solution to the problem can be obtained only by complete enumeration, often at the cost of a prohibitively computation time requirement for realistic power systems. This paper presents a hybrid algorithm which combines Lagrangian Relaxation (LR) together with Evolutionary Algorithm (EA) to solve the problem in cooperative and competitive energy environments. Simulation studies were carried out on different systems containing various numbers of units. The outcomes from different algorithms are compared with that from the proposed hybrid algorithm and the advantages of the proposed algorithm are briefly discussed. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Daily load patterns of many utilities exhibit variation between peak and off-peak hours due to less electricity usage on weekends than on weekdays, and at a lower rate between midnight and early morning than during the day. If sufficient generation is kept online throughout the day, some of the units will be operating near to their minimum generating limits during the off-peak period. This is not an economical solution. Therefore, the problem confronting the system operator is to determine which units should be taken offline during the off-peak period, and which units should be kept online. In most of the interconnected power systems, the power requirement is principally met by thermal power generation. Several operating strategies are possible to meet the required power demand, which varies from hour to hour throughout the day. Moreover, in order to supply high-quality electric power to customers in a secure and economic manner, Unit Commitment (UC) [1] is considered to be one of the best available options. It is thus recognized that the optimal unit commitment, which is the problem of determining the schedule of generating units within a power system, subject to device and operating constraints results ⇑ Corresponding author. Tel.: +65 65500145. E-mail address: [email protected] (T. Logenthiran). http://dx.doi.org/10.1016/j.ijepes.2014.07.044 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
in substantial savings for electric utilities. The general objective of the unit commitment problem is to minimize total operating cost of system while satisfying all of the constraints so that a given security level can be met. Unit commitment problem is one of two linked optimization tasks of generation scheduling problem, which decides ON/OFF statuses of generators over the scheduling period. Economic Dispatch (ED) problem is the other linked optimization problem that finds operating power levels of the committed generators. The electric power industry has been using several efficient methodologies [2] for many years to solve the generation scheduling problem. However, as the power industry undergoes restructuring [3], the role of generation scheduling models is changing. Meanwhile, in modern restructured environments, small improvements in solutions can result significantly in the electricity market [3,4]. Further, deregulation of power systems has motivated small sized production units and distribution generation. As a result, modern power system contains a large number of generating units. As the number of units increases, the generation scheduling problem grows exponentially and needs excessive computation time and effort to solve. The solution to the generation scheduling problem can be obtained only by complete enumeration, often at the cost of a prohibitively large computation time requirements for medium sized power systems. Therefore, research endeavours
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Nomenclature ai,bi & ci C CBUC DRi EA ED Fi(t) Fi(Pi(t)) Fmax Fr Ii(t) J K LR LREA N PBUC Pi(t) Pmax ðtÞ i Pmin ðtÞ i Pl(t) PR(t)
fuel cost coefficients of unit i operating cost cost based unit commitment ramp down rate of generation unit i evolutionary algorithm economic dispatch profit of generation unit i at time t fuel cost of unit i at time t maximum of [Fr] inverse of relative duality gap ON/OFF status of generation unit i at time t (ON = 1 and OFF = 0) primal value of Lagrangian function a scaling factor Lagrangian relaxation Lagrangian relaxation – evolutionary algorithm number of generation units profit-based unit commitment power generation of unit i at time t maximum generation limit of generation unit i at time t minimum generation limit of generation unit i at time t system electricity demand at time t system spinning reserve at time t
have been focused to develop near-optimal generation scheduling algorithms which can be applicable to modern power systems. A literature survey on the unit commitment methodologies reveals that various numerical optimization techniques have been employed to the problem. Specifically, there are traditional mathematical methodologies such as Priority List (PL) [2,3], Dynamic Programming (DP) [4], Branch and Bound (BB) [5] methods, Mixed-Integer Programming (MIP) [6] and Lagrangian Relaxation (LR) [7,8] methods. Among these, priority list is a simple and fast method but the quality of the solution is generally poor. Dynamic programming is flexible but the main disadvantage is the curse of dimensionality which increases in computational time. The shortcoming of branch and bound method is the exponential growth in the execution time with the size of unit commitment problem. Mixed-integer programming method for solving unit commitment problem fails when the number of units increases because that requires a large memory and suffers from great computational delay. Lagrangian relaxation method is used widely to solve unit commitment problem in the power industry. Even though Lagrangian relaxation method provides a fast solution, it may suffer from numerical convergence and solution quality. Recently, solution methodologies based on meta-heuristics such as Genetic Algorithm (GA) [9–12], Evolutionary Programming (EP) [13], Fuzzy Logic (FL), Artificial Neural Network (ANN), Simulated Annealing (SA) and swarm algorithms: Particle Swarm Optimization (PSO) [14–17] and Ant Colony Optimization (ACO) have also shown more promising results. These meta-heuristic optimization methodologies attract much attention because of their ability to search not only local optimal solutions but also global optimal solution and their ability to deal with various difficult nonlinear constraints. However, these meta-heuristic methods require a considerable amount of computational time to find the near-global optimum solution especially for large-scale unit commitment problems. Very recently, researchers show potential of solving with hybrid techniques [18-20]. These techniques that are combinations of the above methodologies, give even better solutions than that from above methodologies. Even though, Lagrangian relaxation method is a widely accepted algorithm in real power system scheduling, and can
q r Re Ri(t) RP(t) SP(t) STci SThi STi(t) T Ti,cold Ti,Down Ti,off(t) Ti,on(t) Ti,up TP UC URi e kt & lt
dual value of Lagrangian function uncertainty in reserve or estimated probability of called reserve revenue allocated reserve power of generation unit i at time t forecasted spinning reserve price at time t forecasted spot market price at time t cold start-up cost hot start-up cost start-up cost of generation unit i at time t number of scheduling time slots in the scheduling period cold start-up time of generation unit i minimum down-time of generation unit i Continuous off-time of generation unit i at time t continuously on-time of generation unit i at time t minimum up time of generation unit i total profit unit commitment ramp-up rate of generation unit i relative duality gap Lagrangian multipliers
provide a fast solution, quality of solutions strongly depends on the algorithm used to update the Lagrangian multipliers. Typically, Lagrangian multipliers are updated by sub-gradient methods with a scaling factor. This paper presents a hybrid algorithm, LREA which uses Lagrangian relaxation together with evolutionary algorithm for solving short-term generation scheduling in cooperative and competitive environments. This hybrid methodology overcomes well-known several disadvantages of traditional sub-gradient methods and other disadvantages of modern computational algorithms such as large simulation time. The remaining paper is organized as follows. Section ‘Problem formulation’ formulates the generation scheduling problem in both cooperative and competitive environments. Section ‘Proposed methodology: Lagrangian Relaxation Hybrids with Evolutionary Algorithm (LREA)’ proposes LREA for solving the generation scheduling problems. Section ‘Simulation studies and results’ provide the details of the simulation studies and simulation outcomes. Section ‘Conclusion’ concludes the paper. Problem formulation Generation scheduling problem involves the determination of states of power generating units for each time slot and power settings of the committed generating units subject to system constraints and generating unit constraints. Objective of generation scheduling problems varies based on the energy environments. This paper studies generation scheduling problems in both cooperative and competitive environments. Cost-based unit commitment problem Traditionally, in a vertically integrated utility which operates in cooperative energy environment, the system operator who has the knowledge of system components, constraints and operating costs of generating units makes decisions on generation scheduling. Cost-Based Unit Commitment [7,21] determines generating unit schedules for a utility by minimizing the operating cost and satisfying the prevailing system and units constraints over the
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scheduling period. This unit commitment problem can be mathematically formulated as follows.
Minimize Total Operating Cost ðTOCÞ ¼
N X T X ½Ii ðtÞ F i ðP i ðtÞÞ
or
Minimize N X T X F i ðtÞ
ð1Þ
ð11Þ
i¼1 t¼1
i¼1 t¼1
The revenue (Re) from the spot energy and reserve is given by the following equation.
þST i ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ Fuel cost of unit i at time t is given by the following equation:
F i ðPi ðtÞÞ ¼ ai þ bi P i ðtÞ þ ci Pi ðtÞ2
ð2Þ
Start-up cost of generation unit i at time t is given by following equation.
ST i ðtÞ ¼
ST hi
if
T i;off ðtÞ T i;Down þ T i;cold
ST ci
if
T i;off ðtÞ > T i;Down þ T i;cold
ð3Þ
This minimization problem is subject to following system and unit constraints. 1. System real power balance N X Ii ðtÞ Pi ðtÞ ¼ Pl ðtÞ
ð4Þ
i¼1
2. System spinning reserve requirement N X Ii ðtÞ Pmax P l ðtÞ þ PR ðtÞ i
ð5Þ
i¼1
Pi ðtÞ
Pmax ðtÞ i
ð6Þ
ð7Þ
5. Ramp up and ramp down rates
Pi ðtÞ P i ðt 1Þ URi Pi ðt 1Þ Pi ðtÞ DRi
ð8Þ
Profit-based unit commitment problem In a restructured system [22,23], generation companies sell power in power market and sell reserve in the ancillary market. These markets provide competitive energy environments for utilities and customers. The amount of power and reserve sold depends on the way reserve payments are made. There are several ways available for reserve payment method in the world such as payment for power delivered, and payment for reserve allocated. In payment for power delivered, reserve is paid when only reserve is actually used. Therefore, the reserve price is higher than the power spot price; whereas, in payment for reserve allocated, generation companies receive the reserve price per unit of reserve for every time period that the reserve is allocated and not used. When the reserve is used, generation companies receive the spot price for the reserve that is generated. In this method, reserve price is much lower than the spot price. In this paper, the power and reserve are sold based on the way reserve payment for power delivered [22]. In this case, profit based generation scheduling problem handled by generation companies under a competitive environment can be formulated mathematically as follows.
Total Profit ðTPÞ ¼ Revenue ðReÞ Operating Cost ðCÞ Maximize N X T X TP ¼ F i ðtÞ i¼1 t¼1
i¼1 t¼1
ð12Þ
i¼1 t¼1
The production cost and start-up cost (C) is calculated by the following equation.
C ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ þ Ri ðtÞÞ i¼1 t¼1
i¼1 t¼1
Ii ðtÞ þ ST i :ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð13Þ
This optimization problem is subject to following system and unit constraints. 1. System energy constraints, N X Pi ðtÞ Ii ðtÞ Pl ðtÞ
ð14Þ
i¼1
N X Ri ðtÞ Ii ðtÞ PR ðtÞ
ð15Þ
i¼1
4. Minimum up and minimum down times
ðT i;on ðt 1Þ T i;Up Þ ðIi ðt 1Þ Ii ðtÞÞ 0 ðT i;off ðt 1Þ T i;Down Þ ðIi ðt 1Þ Ii ðtÞÞ 0
N X T N X T X X Pi ðtÞ SPðtÞ Ii ðtÞ þ r RPðtÞ:Ri ðtÞ Ii ðtÞ
2. System reserve constraints,
3. Generation unit’s limits
Pmin ðtÞ i
Re ¼
ð9Þ
ð10Þ
3. Unit power and reserve limits,
0 Ri ðtÞ Pmax ðtÞ Pmin ðtÞ i i
ð16Þ
Ri ðtÞ þ Pi ðtÞ P max ðtÞ i
ð17Þ
4. Unit minimum UP/Down durations 5. Unit ramping constraints In restructured power markets [22,23], generation companies sell power in the spot market and sell spinning and non-spinning reserves in the reserve markets. As explained above, reserve payments can be made in different scenarios. In this paper, reserve is paid only when actually used. Therefore, the reserve price is higher than the spot price. The probability that the reserves are called and generated is also considered in the formulation to represent the uncertainty in reserves. Further, it is considered that there is no non-spinning reserve in the reserve market. It can be included in the problem formulation easily with another uncertainty constant with it. If the power and reserve are sold based on the way payment for reserve allocated, the formulation of profit based generation scheduling problem will be different in Eqs. (12) and (13). These equations will be replaced by Eqs. (18) and (19) respectively. The revenue (Re) from the spot energy and reserve is given by the following equation.
Re ¼
N X T N X T X X Pi ðtÞ SPðtÞ Ii ðtÞ þ ðð1 rÞ RPðtÞ þ r i¼1 t¼1
i¼1 t¼1
SPðtÞÞ Ri ðtÞ Ii ðtÞ
ð18Þ
The production cost and start-up cost (C) can be calculated by the following equation.
C ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ i¼1 t¼1
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð19Þ
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Proposed methodology: Lagrangian Relaxation Hybrids with Evolutionary Algorithm (LREA) The principal advantage of applying Lagrangian relaxation is its computational efficiency. The execution time of Lagrangian relaxation will increase linearly with the size of the problem [1,7,8]. The Lagrangian relaxation method allows the utilization of parallel computing techniques for single unit commitment sub problems with a small CPU time. However, Lagrangian relaxation decomposition procedure is dependent on the initial estimates of the Lagrangian multipliers, and on the method used to update the multipliers. In order to obtain a near-optimal solution, Lagrange multiplier adjustments are needed to be managed skilfully. In addition, the Lagrangian relaxation method often encounters difficulties as more complicated constraints are considered. The inclusion of a large number of multipliers could result in an optimization problem that is more difficult and even impossible to solve as the number of constraints grows and various heuristics are embedded in the Lagrangian relaxation algorithm. Evolutionary Algorithm (EA) [9–11] is a general purpose stochastic and parallel search method, which can be used as an optimization technique for obtaining near-global optimum solutions of generation scheduling problems. This algorithm is inspired from genetics and evolution theories of natural selection and survival of the fittest. It is iterative procedure acting on a population of chromosomes, each chromosome being the encoding of a candidate solution to the problem. A fitness which depends on how well it solves the problem is associated with each chromosome. The objective function involves penalty term to penalize those potential solutions in which the problem constraints are not fulfilled. The objective function translates into a fitness which determines the solution’s ability to survive and produce offspring. New generations of solutions are obtained by a process of selection, cross-over, and mutation. During the evolution process, new generations should give increasingly fitter solution and evolve toward an optimal solution. In this paper, a hybrid algorithm (i.e. similar to the papers [22,23]) which is combined Lagrangian Relaxation (LR) and Evolutionary Algorithm (EA) is proposed to solve the concern problem. This methodology incorporates evolutionary algorithms into Lagrangian relaxation to update the Lagrangian multipliers and improve the performance of Lagrangian relaxation method. The evolutionary algorithms combine the adaptive nature of the natural genetics or the evolution procedures of organs with functional optimizations. The proposed LREA method consists of a two-stage cycle. The first stage is to search for the constrained minimum of Lagrangian function under constant Lagrangian multipliers by two-state dynamic programming. The second stage is to maximize the Lagrangian function with respect to the Lagrangian multipliers adjusted by evolutionary algorithms. The overall procedure of the proposed method is described in Fig. 1. In the proposed methodology, updating of the Lagrange multipliers is done by an evolutionary algorithm. Components of the evolutionary algorithm are described as follows. A population of chromosomes was constructed as below, and initialised uniform randomly. For T time slots in the scheduling periods, an array of control variable k and l vectors are represented as genes in the chromosomes. This can be shown as below.
k ¼ ½k1 k2 . . . . . . kT l ¼ l1 l2 . . . . . . lT
Fitness 1þK
F max Fr
1
where,
Fr ¼
e¼
1
ð22Þ
e
Jq q
ð23Þ
During the simulation of the algorithm, new population of chromosomes is produced from the existing population by adding Gaussian random number with zero mean and a predefined standard deviation to each individual. Tournament method is used as the selection technique. The duality gap that is the difference between primal and dual problem is used as a terminating criteria for the algorithm. Except the methodology for updating of Lagrange multipliers, the proposed methodology is the same as the traditional Lagrangian relaxation methodology. This is given in the following section. Lagrangian relaxation for generation scheduling The Lagrangian function for generation scheduling in a cooperative energy environment can be formulated as follows.
ð20Þ
Fitness function for evolutionary algorithm is chosen as follows.
1
Fig. 1. LREA for unit commitment problem.
ð21Þ
LðP; R; k; lÞ ¼ JðP; R; IÞ þ
T X t¼1
þ
T X t¼1
lt
kt P l ðtÞ
N X Pi ðtÞ Ii ðtÞ
!
i¼1
N X Pl ðtÞ þ PR ðtÞ P max ðtÞ Ii ðtÞ i i¼1
! ð24Þ
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where
JðP; R; IÞ ¼
N X T X
½Ii ðtÞ F i ðPi ðtÞÞ þ Si ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð25Þ
The Lagrange function can be rewritten by dropping the constant terms as follows. For cooperative environment,
" # N T X X F i ðP i ðtÞÞ þ ST i ð1 Ii ðt 1ÞÞ kt Pi ðtÞ lt Ri ðtÞ Ii ðtÞ
i¼1 t¼1
L¼
Then,
i¼1
t¼1
N X T X LðP; R; k; lÞ ¼ ½Ii ðtÞ F i ðPi ðtÞÞ þ Si ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ i¼1 t¼1 T N X X þ kt Pl ðtÞ Pi ðtÞ Ii ðtÞ t¼1
þ
For competitive environment,
!
i¼1
T X
N X
t¼1
i¼1
lt Pl ðtÞ þ PR ðtÞ
ð32Þ
L¼
!
i¼1
Pmax ðtÞ Ii ðtÞ i ð26Þ
In case of a competitive energy environment, the Lagrangian function for generation scheduling can be formulated as follows.
LðP; R; k; lÞ ¼ JðP; R; IÞ þ
T N X X kt Pl ðtÞ Pi ðtÞ Ii ðtÞ t¼1
þ
T X
lt
i¼1
N X SR Ri ðtÞ Ii ðtÞ
t¼1
!
! ð27Þ
i¼1
where
JðP; R; IÞ ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ i¼1 t¼1
" N T X X
fð1 rÞ F i ðPi ðtÞÞ þ r F i ðP i ðtÞ þ Ri ðtÞÞ
t¼1
þST i ð1 Ii ðt 1ÞÞ P i ðtÞ SPðtÞ r RPðtÞ Ri ðtÞ # þkt Pi ðtÞ þ lt Ri ðtÞ Ii ðtÞ
Term inside the bracket of Eqs. (31) or (32) can be solved separately for each generating unit without regard for what is happening on the other generating units. The minimum of the Lagrange function is found by solving for the minimum for each generating unit over all time periods. Dynamic programming is used to solve this problem. The dynamic programming determines the optimal schedule of each unit over the scheduling time period. More specifically, for each state in each hour, the ON/OFF decision making is needed to select a solution that would result in a higher profit/lower operating cost by comparing the combination of the start-up cost and the accumulated
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ Ii ðtÞ
N X T N X T X X ðPi ðtÞ SPðtÞÞ Ii ðtÞ r RPðtÞ Ri ðtÞ i¼1 t¼1
i¼1 t¼1
Ii ðtÞ
ð28Þ
Then,
LðP; R; k; lÞ ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðP i ðtÞ i¼1 t¼1
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ:Ii ðtÞ
N X T N X T X X ðPi ðtÞ:SPðtÞÞ Ii ðtÞ r RPðtÞ i¼1 t¼1
i¼1 t¼1
T N X X Ri ðtÞ:Ii ðtÞ þ kt Pl ðtÞt Pi ðtÞ Ii ðtÞ t¼1
þ
T X
lt
t¼1
!
i¼1
N X SRðtÞ Ri ðtÞ Ii ðtÞ
! ð29Þ
i¼1
The Lagrangian relaxation procedure solves the unit commitment problem by ‘‘relaxing’’ or temporarily ignoring the coupling constraints and solving the problem as if they did not exist. This is done through the dual optimization procedure which attempts to reach the constrained optimum by maximizing the Lagrangian function with respect to the Lagrangian multipliers and minimizing the function with respect to the control variables I, P, and R.
q ¼ max qðk; lÞ k;l
ð30Þ
where
qðk; lÞ ¼ min LðP; R; k; l; IÞ I;P;R
ð31Þ
Subject to the constraints given in the problem formations. Fig. 2 shows a flowchart of Lagrangian relaxation methodology for unit commitment problems. The detail of the algorithm is described step by step as in details as follows.
ð33Þ
Fig. 2. Lagrangian relaxation for unit commitment problem.
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Fig. 3. Search paths of each unit in dynamic programming.
profits from two historical routes. This procedure is shown in Fig. 3. At this stage calculate the dual value of the problem. Then, in order to maximize the Lagrange function, Lagrange multipliers must be suitably adjusted. Traditionally, it is done by a gradient
based approach. Once the new Lagrange multipliers are known, power and reserve settings of each committed units can be found by Economic Dispatch (ED) [1]. The economic dispatch of generators is a key element in the optimal operation of power generation systems. The main goal is the generation of a given amount of electricity at the highest profit/least operating cost possible. Once the unit commitment status is determined, an economic dispatch problem is formulated and solved to ensure the feasibility of the original unit commitment solution. The economic dispatch problem at time is given as follows. For cooperative environment
Minimize N X JðI; PÞ
ð34Þ
i¼1
Table 1 Data of 10-unit system. Parameters
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
Pmax (MW) Pmin (MW) Rmax (MW) Rmin (MW) a ($/h) b ($/MW h) c ($/MW2 h) 10–4 Min up time (h) Min down time (h) Hot start-up cost ($) Cold start-up cost ($) Cold start-up hrs (h) Initial status (h)
455 150 45.5 0 1000 16.19 4.8 8 8 4500 9000 5 8
455 150 45.5 0 970 17.26 3.1 8 8 5000 10,000 5 8
130 20 13.0 0 700 16.60 20 5 5 550 1100 4 5
130 20 13.0 0 680 16.50 21.1 5 5 560 1120 4 5
162 25 16.2 0 450 19.70 39.8 6 6 900 1800 4 6
80 20 8.0 0 370 22.26 71.2 3 3 170 340 2 3
85 25 8.5 0 480 27.74 7.9 3 3 260 520 2 3
55 10 5.5 0 660 25.92 41.3 1 1 30 60 0 1
55 10 5.5 0 665 27.27 22.2 1 1 30 60 0 1
55 10 5.5 0 670 27.79 17.3 1 1 30 60 0 1
Table 2 Forecasted demands, spinning reserves and spot market prices for 10-unit system. Hour
Demand (MW)
Reserve (MW)
Spot price ($/MW h)
Hour
Demand (MW)
Reserve (MW)
Spot price ($/MW h)
1 2 3 4 5 6 7 8 9 10 11 12
700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500
70 75 85 95 100 110 115 120 130 140 145 150
22.15 22.00 23.10 22.65 23.25 22.95 22.50 22.15 22.80 29.35 30.15 31.65
13 14 15 16 17 18 19 20 21 22 23 24
1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800
140 130 120 105 100 110 120 140 130 110 90 80
24.60 24.50 22.50 22.30 22.25 22.05 22.20 22.65 23.10 22.95 22.75 22.55
Table 3 Simulation results obtained for cooperative energy environment. No of units
10 20 40 60 80 100 150 200 300 400 500
GA
LR
LREA
Average cost ($)
Elapsed time (s)
Average cost ($)
Elapsed time (s)
Average cost ($)
Elapsed time (s)
565,825 1,126,249 2,251,981 3,383,631 4,504,989 5,587,538 8,405,341 11,175,072 16,762,600 22,350,152 27,937,680
823 1443 2411 3912 5181 8348 10,750 16,690 24,043 33,081 40,001
565,825 1,120,660 2,258,510 3,394,056 4,526,043 5,657,290 8,667,823 11,304,567 17,002,344 23,000,234 29,023,445
117 210 382 592 890 1312 1603 2102 3201 4322 5023
565,825 1,120,660 2,248,700 3,367,902 4,492,012 5,510,320 8,400,123 11,110,343 16,053,852 21,943,688 26,707,993
1232 1301 1352 1501 1981 2568 3023 4234 5983 6502 7034
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Table 4 Power setting of each unit of 10 unit system in cooperative energy environment. Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Power settings (MW) Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455
245 295 370 455 390 360 410 455 455 455 455 455 455 455 455 310 260 360 455 455 455 455 420 345
0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 25 40 25 25 25 30 85 162 162 162 162 85 30 25 25 25 30 162 85 145 25 0
0 0 0 0 0 0 0 0 20 33 73 80 33 20 0 0 0 0 0 33 20 20 0 0
0 0 0 0 0 0 0 0 25 25 25 25 25 25 0 0 0 0 0 25 25 25 0 0
0 0 0 0 0 0 0 0 0 10 10 43 10 0 0 0 0 0 0 10 0 0 0 0
0 0 0 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0
Table 5 Power setting of each unit of 10 unit system in competitive energy environment. Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Power settings (MW)/reserve settings (MW) Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0
245/70 295/75 395/60 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 415/40 455/0 455/0 455/0 455/0 455/0 445/10 345/80
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 162/0 162/0 162/0 162/0 130/32 160/2 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 68/12 80/0 80/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
For competitive environment
Minimize N X F i ðtÞ
Simulation studies and results
ð35Þ
i¼1
Subject to energy requirement, reserve requirement and generation unit limits. Typical Lambda-iteration method [1] is used to solve the economic dispatch problem. The iteration continuous until it meets the convergence criteria. Here the relative duality gap is used as the convergence criteria.
Relative Duality Gap ¼
Jq q
ð36Þ
Initially, some simulation studies [21,22] were carried out on test systems for benchmarking of the proposed algorithm. Then the feasibility of the proposed algorithm is demonstrated on power systems with much larger number of units. Cooperative environment The feasibility of the proposed algorithms is demonstrated with different size of power systems containing 10–500 units and the test results are compared with each other in terms of solution quality and convergence characteristic. Characteristics of units and load for the 10-units system are given in Tables 1 and 2.
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T. Logenthiran et al. / Electrical Power and Energy Systems 64 (2015) 356–364 Table 6 Profits obtained for competitive energy environment. No of units
10 20 40 60 80 100 150 200 300 400 500
GA
LR
LREA
Average profit ($)
Elapsed time (s)
Average profit ($)
Elapsed time (s)
Average profit ($)
Elapsed time (s)
107,872 215,621 431,361 644,676 837,729 1,038,262 1,577,493 2,075,654 3,115,765 4,156,760 5,209,568
752 1323 2321 3821 5099 8235 11,243 16,345 23,654 31,456 42,497
107,872 215,743 431,223 642,122 812,499 997,924 1,528,782 2,000,234 3,109,456 4,098,234 5,198,374
111 198 368 571 851 1298 2046 3294 4594 5987 7943
107,875 215,747 431,489 647,235 862,979 1,078,718 1,678,332 2,164,749 3,278,398 4,387,573 5,457,975
1112 1230 1365 1498 1954 2537 3021 4364 5868 7365 9023
Spinning reserve of the system is assumed to be 10% of the load. The different number of units systems is created by repeating each unit of the original problem by respective factors. Thus, test problems with different numbers units are obtained. Hourly load and reserve amounts are also scaled by the same factors. After proper tuning the parameters of the proposed algorithm, simulation studies were carried out. The best results obtained by the algorithm are tabulated in Table 3. Further, Table 4 shows the power settings of each unit in case of 10 unit system when LREA is used. The proposed algorithm provides better numerical convergence for all the systems considered in the study. Currently, LR method is a widely accepted algorithm in real power system scheduling which is considered as a benchmark for comparing the results. It is shown in the paper that LREA provides better solutions. Competitive environment Similar to the cooperative environment, the feasibility of the proposed algorithms is demonstrated in competitive environment with different size of power systems containing 10–500 units. The same ten unit system is used in these studies also. Forecasted loads, spot prices, and reserve prices for the system are given in Table 2. In these studies, reserve prices are fixed at five times the spot prices, and probability that reserve is called and generated is kept at 0.05. Simulations were carried out and $113134.12 profit is achieved by the proposed algorithm. It is higher than that from [22]. The corresponding power settings of each unit are given in Table 5. Lagrangian relaxation and genetic algorithm are also used for generation scheduling of the same systems and used as benchmark algorithms. The best results obtained by the algorithm are tabulated in Table 6 for other systems containing different number of units. The proposed LREA algorithm provides better numerical convergence for all the systems considered in the simulation, and provides better solutions. LR for unit commitment problem provides a fast solution but it may suffer from numerical convergence and solution quality and Evolutionary Algorithm (EA) methods find the better solution but they need more computational time. In addition, LR decomposition procedure is dependent on the initial estimates of the Lagrangian multipliers and on the method used to update the multipliers. Currently the techniques used for estimating the Lagrangian multipliers rely on a sub-gradient algorithm or heuristics. Incorporating evolutionary algorithm into LR method to update the Lagrangian multipliers improves the performance of LR method in solving the unit commitment problem. As the result, it provides better numerical results obtained in a reasonable computational time.
Conclusion This paper proposes a hybrid algorithm to solve the short-term generation scheduling problem with operational constraints in a traditional and a restructured power systems using Lagrangian relaxation and evolutionary algorithm. In the restructured power system, spot prices and reserve prices in the market are important parameters while solving the profit based unit commitment problem. Using the proposed LREA approach, the generation companies can maximize their profit and schedule the generating units accordingly. The results achieved are quite encouraging and indicate the feasibility of the proposed technique to deal with large unit commitment problems in a deregulated environment. Furthermore, the proposed algorithm can find the most economical scheduling plan for traditional power systems according to the results. Using the proposed approach, the system can minimize the operating cost of generators substantially. According to the numerical results, LREA seems to be the best algorithm among the algorithms discussed in this paper for short-term generation scheduling problems in both cooperative and competitive energy environments.
References [1] Wood J, Wollenberg BF. Power generation, operation & control. John Wiley & Sons, Inc.; 1996. [2] Burns RM, Gibson CA. Optimization of priority lists for a unit commitment program. IEEE PES Summer Meet 1975. [3] Senjyu T, Miyagi T, Saber A. Emerging solution of large-scale unit commitment problem by stochastic priority list. J Electr Power Syst Res 2006:283–92. [4] Ouyang Z, Shahidehpour SM. An intelligent dynamic programming for unit commitment application. IEEE Trans Power Syst 1991:1203–9. [5] Cohen AI, Yoshimura M. A branch-and-bound algorithm for unit commitment. IEEE Trans Power Apparatus Syst 1983:444–51. [6] Carrión M, Arroyo JM. A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans Power Syst 2006;21(3):1371–8. [7] Wang SJ, Shahidehpour SM, Kirschen DS, S.Mokhtari, Irisarri GD. Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation. IEEE Trans Power Syst 1995:1294–301. [8] Ongsakul W, Petcharaks N. Unit commitment by enhanced adaptive Lagrangian relaxation. IEEE Trans Power Syst 2004:620–8. [9] Kazarlis SA, Bakirtzis AG, Petridis V. A genetic algorithm solution to the unit commitment problem. IEEE Trans Power Syst 1996:83–92. [10] Dasgupta D, McGregor DR. Thermal unit commitment using genetic algorithms. IEE Proc C, Gener Transmission Distribution 1994:459–65. [11] Sheble GB, Maifeld TT. Unit commitment by genetic algorithm and expert system. J Electr Power Syst Res 1994:115–21. [12] Srinivasan D, Chazelas J. A priority list-based evolutionary algorithm to solve large scale unit commitment problem. IEEE PowerCon 2004:1746–51. [13] Juste KA, Kita H, Tanaka E, Hasegawa J. An evolutionary programming solution to the unit commitment problem. IEEE Trans Power Syst 1999:1452–9. [14] Zwe LG. Discrete particle swarm optimization algorithm for unit commitment. IEEE PES Gen Meet 2003:418–24. [15] Yuan X, Nie H, Su A, Wang L, Yuan Y. An improved binary particle swarm optimization for unit commitment problem. J Expert Syst Appl 2009:8049–55.
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[16] Saber AY, Senjyu T, Urasaki N, Funabashi T. Unit commitment computation – a novel fuzzy adaptive particle swarm optimization approach power systems conference and exposition. IEEE PSCE 2006:1820–8. [17] Ting T, Rao M, Loo C. Solving unit commitment problem using hybrid particle swarm optimization. J Heuristics 2006:507–20. [18] Cheng CP, Liu CW, Liu CC. Unit commitment by Lagrangian relaxation and genetic algorithms. IEEE Trans Power Syst 2000:707–14. [19] Balci H, Valenzuela J. Scheduling electric power generations using particle swarm optimization combined with the lagrangian relaxation method. J Appl Mathematics Comput Sci 2004:411–21.
[20] Logenthiran T, Srinivasan D. Short term generation scheduling of a microgrid. IEEE TENCON 2009:1–6. [21] Logenthiran T, Srinivasan D. Particle swarm optimization for unit commitment problem. IEEE Int Conf Probabilistic Methods Appl Power Syst (PMAPS). 2010:642–7. [22] Logenthiran T, Srinivasan D. LRGA for solving profit based generation scheduling problem in competitive environment. IEEE Congr Evol Comput (CEC) 2011:1148–54. [23] Attavinyanupap P, Kita H, Tanaka E, Hasegawa J. A Hybrid LR-EP for salving new profit-based UC problem under competitive environment. IEEE Trans Power Syst 2003:229–37.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Lagrangian relaxation hybrid with evolutionary algorithm for short-term generation scheduling Thillainathan Logenthiran ⇑, Wai Lok Woo, Van Tung Phan School of Electrical and Electronic Engineering, Newcastle University, Singapore Campus, 180 Ang Mo Kio Avenue 8, Block P, Level 2, Unit 220 (Room P.220B), Singapore 569830, Singapore
a r t i c l e
i n f o
Article history: Received 14 November 2013 Received in revised form 30 June 2014 Accepted 17 July 2014
Keywords: Short-term generation scheduling Profit-based unit commitment Cost-based unit commitment Lagrangian relaxation Evolutionary algorithm Economic dispatch
a b s t r a c t Short-term generation scheduling is an important function in daily operational planning of power systems. It is defined as optimal scheduling of power generators over a scheduling period while respecting various generator constraints and system constraints. Objective of the problem includes costs associated with energy production, start-up cost and shut-down cost along with profits. The resulting problem is a large scale nonlinear mixed-integer optimization problem for which there is no exact solution technique available. The solution to the problem can be obtained only by complete enumeration, often at the cost of a prohibitively computation time requirement for realistic power systems. This paper presents a hybrid algorithm which combines Lagrangian Relaxation (LR) together with Evolutionary Algorithm (EA) to solve the problem in cooperative and competitive energy environments. Simulation studies were carried out on different systems containing various numbers of units. The outcomes from different algorithms are compared with that from the proposed hybrid algorithm and the advantages of the proposed algorithm are briefly discussed. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Daily load patterns of many utilities exhibit variation between peak and off-peak hours due to less electricity usage on weekends than on weekdays, and at a lower rate between midnight and early morning than during the day. If sufficient generation is kept online throughout the day, some of the units will be operating near to their minimum generating limits during the off-peak period. This is not an economical solution. Therefore, the problem confronting the system operator is to determine which units should be taken offline during the off-peak period, and which units should be kept online. In most of the interconnected power systems, the power requirement is principally met by thermal power generation. Several operating strategies are possible to meet the required power demand, which varies from hour to hour throughout the day. Moreover, in order to supply high-quality electric power to customers in a secure and economic manner, Unit Commitment (UC) [1] is considered to be one of the best available options. It is thus recognized that the optimal unit commitment, which is the problem of determining the schedule of generating units within a power system, subject to device and operating constraints results ⇑ Corresponding author. Tel.: +65 65500145. E-mail address: [email protected] (T. Logenthiran). http://dx.doi.org/10.1016/j.ijepes.2014.07.044 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
in substantial savings for electric utilities. The general objective of the unit commitment problem is to minimize total operating cost of system while satisfying all of the constraints so that a given security level can be met. Unit commitment problem is one of two linked optimization tasks of generation scheduling problem, which decides ON/OFF statuses of generators over the scheduling period. Economic Dispatch (ED) problem is the other linked optimization problem that finds operating power levels of the committed generators. The electric power industry has been using several efficient methodologies [2] for many years to solve the generation scheduling problem. However, as the power industry undergoes restructuring [3], the role of generation scheduling models is changing. Meanwhile, in modern restructured environments, small improvements in solutions can result significantly in the electricity market [3,4]. Further, deregulation of power systems has motivated small sized production units and distribution generation. As a result, modern power system contains a large number of generating units. As the number of units increases, the generation scheduling problem grows exponentially and needs excessive computation time and effort to solve. The solution to the generation scheduling problem can be obtained only by complete enumeration, often at the cost of a prohibitively large computation time requirements for medium sized power systems. Therefore, research endeavours
T. Logenthiran et al. / Electrical Power and Energy Systems 64 (2015) 356–364
357
Nomenclature ai,bi & ci C CBUC DRi EA ED Fi(t) Fi(Pi(t)) Fmax Fr Ii(t) J K LR LREA N PBUC Pi(t) Pmax ðtÞ i Pmin ðtÞ i Pl(t) PR(t)
fuel cost coefficients of unit i operating cost cost based unit commitment ramp down rate of generation unit i evolutionary algorithm economic dispatch profit of generation unit i at time t fuel cost of unit i at time t maximum of [Fr] inverse of relative duality gap ON/OFF status of generation unit i at time t (ON = 1 and OFF = 0) primal value of Lagrangian function a scaling factor Lagrangian relaxation Lagrangian relaxation – evolutionary algorithm number of generation units profit-based unit commitment power generation of unit i at time t maximum generation limit of generation unit i at time t minimum generation limit of generation unit i at time t system electricity demand at time t system spinning reserve at time t
have been focused to develop near-optimal generation scheduling algorithms which can be applicable to modern power systems. A literature survey on the unit commitment methodologies reveals that various numerical optimization techniques have been employed to the problem. Specifically, there are traditional mathematical methodologies such as Priority List (PL) [2,3], Dynamic Programming (DP) [4], Branch and Bound (BB) [5] methods, Mixed-Integer Programming (MIP) [6] and Lagrangian Relaxation (LR) [7,8] methods. Among these, priority list is a simple and fast method but the quality of the solution is generally poor. Dynamic programming is flexible but the main disadvantage is the curse of dimensionality which increases in computational time. The shortcoming of branch and bound method is the exponential growth in the execution time with the size of unit commitment problem. Mixed-integer programming method for solving unit commitment problem fails when the number of units increases because that requires a large memory and suffers from great computational delay. Lagrangian relaxation method is used widely to solve unit commitment problem in the power industry. Even though Lagrangian relaxation method provides a fast solution, it may suffer from numerical convergence and solution quality. Recently, solution methodologies based on meta-heuristics such as Genetic Algorithm (GA) [9–12], Evolutionary Programming (EP) [13], Fuzzy Logic (FL), Artificial Neural Network (ANN), Simulated Annealing (SA) and swarm algorithms: Particle Swarm Optimization (PSO) [14–17] and Ant Colony Optimization (ACO) have also shown more promising results. These meta-heuristic optimization methodologies attract much attention because of their ability to search not only local optimal solutions but also global optimal solution and their ability to deal with various difficult nonlinear constraints. However, these meta-heuristic methods require a considerable amount of computational time to find the near-global optimum solution especially for large-scale unit commitment problems. Very recently, researchers show potential of solving with hybrid techniques [18-20]. These techniques that are combinations of the above methodologies, give even better solutions than that from above methodologies. Even though, Lagrangian relaxation method is a widely accepted algorithm in real power system scheduling, and can
q r Re Ri(t) RP(t) SP(t) STci SThi STi(t) T Ti,cold Ti,Down Ti,off(t) Ti,on(t) Ti,up TP UC URi e kt & lt
dual value of Lagrangian function uncertainty in reserve or estimated probability of called reserve revenue allocated reserve power of generation unit i at time t forecasted spinning reserve price at time t forecasted spot market price at time t cold start-up cost hot start-up cost start-up cost of generation unit i at time t number of scheduling time slots in the scheduling period cold start-up time of generation unit i minimum down-time of generation unit i Continuous off-time of generation unit i at time t continuously on-time of generation unit i at time t minimum up time of generation unit i total profit unit commitment ramp-up rate of generation unit i relative duality gap Lagrangian multipliers
provide a fast solution, quality of solutions strongly depends on the algorithm used to update the Lagrangian multipliers. Typically, Lagrangian multipliers are updated by sub-gradient methods with a scaling factor. This paper presents a hybrid algorithm, LREA which uses Lagrangian relaxation together with evolutionary algorithm for solving short-term generation scheduling in cooperative and competitive environments. This hybrid methodology overcomes well-known several disadvantages of traditional sub-gradient methods and other disadvantages of modern computational algorithms such as large simulation time. The remaining paper is organized as follows. Section ‘Problem formulation’ formulates the generation scheduling problem in both cooperative and competitive environments. Section ‘Proposed methodology: Lagrangian Relaxation Hybrids with Evolutionary Algorithm (LREA)’ proposes LREA for solving the generation scheduling problems. Section ‘Simulation studies and results’ provide the details of the simulation studies and simulation outcomes. Section ‘Conclusion’ concludes the paper. Problem formulation Generation scheduling problem involves the determination of states of power generating units for each time slot and power settings of the committed generating units subject to system constraints and generating unit constraints. Objective of generation scheduling problems varies based on the energy environments. This paper studies generation scheduling problems in both cooperative and competitive environments. Cost-based unit commitment problem Traditionally, in a vertically integrated utility which operates in cooperative energy environment, the system operator who has the knowledge of system components, constraints and operating costs of generating units makes decisions on generation scheduling. Cost-Based Unit Commitment [7,21] determines generating unit schedules for a utility by minimizing the operating cost and satisfying the prevailing system and units constraints over the
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scheduling period. This unit commitment problem can be mathematically formulated as follows.
Minimize Total Operating Cost ðTOCÞ ¼
N X T X ½Ii ðtÞ F i ðP i ðtÞÞ
or
Minimize N X T X F i ðtÞ
ð1Þ
ð11Þ
i¼1 t¼1
i¼1 t¼1
The revenue (Re) from the spot energy and reserve is given by the following equation.
þST i ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ Fuel cost of unit i at time t is given by the following equation:
F i ðPi ðtÞÞ ¼ ai þ bi P i ðtÞ þ ci Pi ðtÞ2
ð2Þ
Start-up cost of generation unit i at time t is given by following equation.
ST i ðtÞ ¼
ST hi
if
T i;off ðtÞ T i;Down þ T i;cold
ST ci
if
T i;off ðtÞ > T i;Down þ T i;cold
ð3Þ
This minimization problem is subject to following system and unit constraints. 1. System real power balance N X Ii ðtÞ Pi ðtÞ ¼ Pl ðtÞ
ð4Þ
i¼1
2. System spinning reserve requirement N X Ii ðtÞ Pmax P l ðtÞ þ PR ðtÞ i
ð5Þ
i¼1
Pi ðtÞ
Pmax ðtÞ i
ð6Þ
ð7Þ
5. Ramp up and ramp down rates
Pi ðtÞ P i ðt 1Þ URi Pi ðt 1Þ Pi ðtÞ DRi
ð8Þ
Profit-based unit commitment problem In a restructured system [22,23], generation companies sell power in power market and sell reserve in the ancillary market. These markets provide competitive energy environments for utilities and customers. The amount of power and reserve sold depends on the way reserve payments are made. There are several ways available for reserve payment method in the world such as payment for power delivered, and payment for reserve allocated. In payment for power delivered, reserve is paid when only reserve is actually used. Therefore, the reserve price is higher than the power spot price; whereas, in payment for reserve allocated, generation companies receive the reserve price per unit of reserve for every time period that the reserve is allocated and not used. When the reserve is used, generation companies receive the spot price for the reserve that is generated. In this method, reserve price is much lower than the spot price. In this paper, the power and reserve are sold based on the way reserve payment for power delivered [22]. In this case, profit based generation scheduling problem handled by generation companies under a competitive environment can be formulated mathematically as follows.
Total Profit ðTPÞ ¼ Revenue ðReÞ Operating Cost ðCÞ Maximize N X T X TP ¼ F i ðtÞ i¼1 t¼1
i¼1 t¼1
ð12Þ
i¼1 t¼1
The production cost and start-up cost (C) is calculated by the following equation.
C ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ þ Ri ðtÞÞ i¼1 t¼1
i¼1 t¼1
Ii ðtÞ þ ST i :ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð13Þ
This optimization problem is subject to following system and unit constraints. 1. System energy constraints, N X Pi ðtÞ Ii ðtÞ Pl ðtÞ
ð14Þ
i¼1
N X Ri ðtÞ Ii ðtÞ PR ðtÞ
ð15Þ
i¼1
4. Minimum up and minimum down times
ðT i;on ðt 1Þ T i;Up Þ ðIi ðt 1Þ Ii ðtÞÞ 0 ðT i;off ðt 1Þ T i;Down Þ ðIi ðt 1Þ Ii ðtÞÞ 0
N X T N X T X X Pi ðtÞ SPðtÞ Ii ðtÞ þ r RPðtÞ:Ri ðtÞ Ii ðtÞ
2. System reserve constraints,
3. Generation unit’s limits
Pmin ðtÞ i
Re ¼
ð9Þ
ð10Þ
3. Unit power and reserve limits,
0 Ri ðtÞ Pmax ðtÞ Pmin ðtÞ i i
ð16Þ
Ri ðtÞ þ Pi ðtÞ P max ðtÞ i
ð17Þ
4. Unit minimum UP/Down durations 5. Unit ramping constraints In restructured power markets [22,23], generation companies sell power in the spot market and sell spinning and non-spinning reserves in the reserve markets. As explained above, reserve payments can be made in different scenarios. In this paper, reserve is paid only when actually used. Therefore, the reserve price is higher than the spot price. The probability that the reserves are called and generated is also considered in the formulation to represent the uncertainty in reserves. Further, it is considered that there is no non-spinning reserve in the reserve market. It can be included in the problem formulation easily with another uncertainty constant with it. If the power and reserve are sold based on the way payment for reserve allocated, the formulation of profit based generation scheduling problem will be different in Eqs. (12) and (13). These equations will be replaced by Eqs. (18) and (19) respectively. The revenue (Re) from the spot energy and reserve is given by the following equation.
Re ¼
N X T N X T X X Pi ðtÞ SPðtÞ Ii ðtÞ þ ðð1 rÞ RPðtÞ þ r i¼1 t¼1
i¼1 t¼1
SPðtÞÞ Ri ðtÞ Ii ðtÞ
ð18Þ
The production cost and start-up cost (C) can be calculated by the following equation.
C ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ i¼1 t¼1
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð19Þ
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T. Logenthiran et al. / Electrical Power and Energy Systems 64 (2015) 356–364
Proposed methodology: Lagrangian Relaxation Hybrids with Evolutionary Algorithm (LREA) The principal advantage of applying Lagrangian relaxation is its computational efficiency. The execution time of Lagrangian relaxation will increase linearly with the size of the problem [1,7,8]. The Lagrangian relaxation method allows the utilization of parallel computing techniques for single unit commitment sub problems with a small CPU time. However, Lagrangian relaxation decomposition procedure is dependent on the initial estimates of the Lagrangian multipliers, and on the method used to update the multipliers. In order to obtain a near-optimal solution, Lagrange multiplier adjustments are needed to be managed skilfully. In addition, the Lagrangian relaxation method often encounters difficulties as more complicated constraints are considered. The inclusion of a large number of multipliers could result in an optimization problem that is more difficult and even impossible to solve as the number of constraints grows and various heuristics are embedded in the Lagrangian relaxation algorithm. Evolutionary Algorithm (EA) [9–11] is a general purpose stochastic and parallel search method, which can be used as an optimization technique for obtaining near-global optimum solutions of generation scheduling problems. This algorithm is inspired from genetics and evolution theories of natural selection and survival of the fittest. It is iterative procedure acting on a population of chromosomes, each chromosome being the encoding of a candidate solution to the problem. A fitness which depends on how well it solves the problem is associated with each chromosome. The objective function involves penalty term to penalize those potential solutions in which the problem constraints are not fulfilled. The objective function translates into a fitness which determines the solution’s ability to survive and produce offspring. New generations of solutions are obtained by a process of selection, cross-over, and mutation. During the evolution process, new generations should give increasingly fitter solution and evolve toward an optimal solution. In this paper, a hybrid algorithm (i.e. similar to the papers [22,23]) which is combined Lagrangian Relaxation (LR) and Evolutionary Algorithm (EA) is proposed to solve the concern problem. This methodology incorporates evolutionary algorithms into Lagrangian relaxation to update the Lagrangian multipliers and improve the performance of Lagrangian relaxation method. The evolutionary algorithms combine the adaptive nature of the natural genetics or the evolution procedures of organs with functional optimizations. The proposed LREA method consists of a two-stage cycle. The first stage is to search for the constrained minimum of Lagrangian function under constant Lagrangian multipliers by two-state dynamic programming. The second stage is to maximize the Lagrangian function with respect to the Lagrangian multipliers adjusted by evolutionary algorithms. The overall procedure of the proposed method is described in Fig. 1. In the proposed methodology, updating of the Lagrange multipliers is done by an evolutionary algorithm. Components of the evolutionary algorithm are described as follows. A population of chromosomes was constructed as below, and initialised uniform randomly. For T time slots in the scheduling periods, an array of control variable k and l vectors are represented as genes in the chromosomes. This can be shown as below.
k ¼ ½k1 k2 . . . . . . kT l ¼ l1 l2 . . . . . . lT
Fitness 1þK
F max Fr
1
where,
Fr ¼
e¼
1
ð22Þ
e
Jq q
ð23Þ
During the simulation of the algorithm, new population of chromosomes is produced from the existing population by adding Gaussian random number with zero mean and a predefined standard deviation to each individual. Tournament method is used as the selection technique. The duality gap that is the difference between primal and dual problem is used as a terminating criteria for the algorithm. Except the methodology for updating of Lagrange multipliers, the proposed methodology is the same as the traditional Lagrangian relaxation methodology. This is given in the following section. Lagrangian relaxation for generation scheduling The Lagrangian function for generation scheduling in a cooperative energy environment can be formulated as follows.
ð20Þ
Fitness function for evolutionary algorithm is chosen as follows.
1
Fig. 1. LREA for unit commitment problem.
ð21Þ
LðP; R; k; lÞ ¼ JðP; R; IÞ þ
T X t¼1
þ
T X t¼1
lt
kt P l ðtÞ
N X Pi ðtÞ Ii ðtÞ
!
i¼1
N X Pl ðtÞ þ PR ðtÞ P max ðtÞ Ii ðtÞ i i¼1
! ð24Þ
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where
JðP; R; IÞ ¼
N X T X
½Ii ðtÞ F i ðPi ðtÞÞ þ Si ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ
ð25Þ
The Lagrange function can be rewritten by dropping the constant terms as follows. For cooperative environment,
" # N T X X F i ðP i ðtÞÞ þ ST i ð1 Ii ðt 1ÞÞ kt Pi ðtÞ lt Ri ðtÞ Ii ðtÞ
i¼1 t¼1
L¼
Then,
i¼1
t¼1
N X T X LðP; R; k; lÞ ¼ ½Ii ðtÞ F i ðPi ðtÞÞ þ Si ðtÞ ð1 Ii ðt 1ÞÞ Ii ðtÞ i¼1 t¼1 T N X X þ kt Pl ðtÞ Pi ðtÞ Ii ðtÞ t¼1
þ
For competitive environment,
!
i¼1
T X
N X
t¼1
i¼1
lt Pl ðtÞ þ PR ðtÞ
ð32Þ
L¼
!
i¼1
Pmax ðtÞ Ii ðtÞ i ð26Þ
In case of a competitive energy environment, the Lagrangian function for generation scheduling can be formulated as follows.
LðP; R; k; lÞ ¼ JðP; R; IÞ þ
T N X X kt Pl ðtÞ Pi ðtÞ Ii ðtÞ t¼1
þ
T X
lt
i¼1
N X SR Ri ðtÞ Ii ðtÞ
t¼1
!
! ð27Þ
i¼1
where
JðP; R; IÞ ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðPi ðtÞ i¼1 t¼1
" N T X X
fð1 rÞ F i ðPi ðtÞÞ þ r F i ðP i ðtÞ þ Ri ðtÞÞ
t¼1
þST i ð1 Ii ðt 1ÞÞ P i ðtÞ SPðtÞ r RPðtÞ Ri ðtÞ # þkt Pi ðtÞ þ lt Ri ðtÞ Ii ðtÞ
Term inside the bracket of Eqs. (31) or (32) can be solved separately for each generating unit without regard for what is happening on the other generating units. The minimum of the Lagrange function is found by solving for the minimum for each generating unit over all time periods. Dynamic programming is used to solve this problem. The dynamic programming determines the optimal schedule of each unit over the scheduling time period. More specifically, for each state in each hour, the ON/OFF decision making is needed to select a solution that would result in a higher profit/lower operating cost by comparing the combination of the start-up cost and the accumulated
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ Ii ðtÞ
N X T N X T X X ðPi ðtÞ SPðtÞÞ Ii ðtÞ r RPðtÞ Ri ðtÞ i¼1 t¼1
i¼1 t¼1
Ii ðtÞ
ð28Þ
Then,
LðP; R; k; lÞ ¼ ð1 rÞ
N X T N X T X X F i ðPi ðtÞÞ Ii ðtÞ þ r F i ðP i ðtÞ i¼1 t¼1
i¼1 t¼1
þ Ri ðtÞÞ Ii ðtÞ þ ST i ð1 Ii ðt 1ÞÞ:Ii ðtÞ
N X T N X T X X ðPi ðtÞ:SPðtÞÞ Ii ðtÞ r RPðtÞ i¼1 t¼1
i¼1 t¼1
T N X X Ri ðtÞ:Ii ðtÞ þ kt Pl ðtÞt Pi ðtÞ Ii ðtÞ t¼1
þ
T X
lt
t¼1
!
i¼1
N X SRðtÞ Ri ðtÞ Ii ðtÞ
! ð29Þ
i¼1
The Lagrangian relaxation procedure solves the unit commitment problem by ‘‘relaxing’’ or temporarily ignoring the coupling constraints and solving the problem as if they did not exist. This is done through the dual optimization procedure which attempts to reach the constrained optimum by maximizing the Lagrangian function with respect to the Lagrangian multipliers and minimizing the function with respect to the control variables I, P, and R.
q ¼ max qðk; lÞ k;l
ð30Þ
where
qðk; lÞ ¼ min LðP; R; k; l; IÞ I;P;R
ð31Þ
Subject to the constraints given in the problem formations. Fig. 2 shows a flowchart of Lagrangian relaxation methodology for unit commitment problems. The detail of the algorithm is described step by step as in details as follows.
ð33Þ
Fig. 2. Lagrangian relaxation for unit commitment problem.
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Fig. 3. Search paths of each unit in dynamic programming.
profits from two historical routes. This procedure is shown in Fig. 3. At this stage calculate the dual value of the problem. Then, in order to maximize the Lagrange function, Lagrange multipliers must be suitably adjusted. Traditionally, it is done by a gradient
based approach. Once the new Lagrange multipliers are known, power and reserve settings of each committed units can be found by Economic Dispatch (ED) [1]. The economic dispatch of generators is a key element in the optimal operation of power generation systems. The main goal is the generation of a given amount of electricity at the highest profit/least operating cost possible. Once the unit commitment status is determined, an economic dispatch problem is formulated and solved to ensure the feasibility of the original unit commitment solution. The economic dispatch problem at time is given as follows. For cooperative environment
Minimize N X JðI; PÞ
ð34Þ
i¼1
Table 1 Data of 10-unit system. Parameters
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
Pmax (MW) Pmin (MW) Rmax (MW) Rmin (MW) a ($/h) b ($/MW h) c ($/MW2 h) 10–4 Min up time (h) Min down time (h) Hot start-up cost ($) Cold start-up cost ($) Cold start-up hrs (h) Initial status (h)
455 150 45.5 0 1000 16.19 4.8 8 8 4500 9000 5 8
455 150 45.5 0 970 17.26 3.1 8 8 5000 10,000 5 8
130 20 13.0 0 700 16.60 20 5 5 550 1100 4 5
130 20 13.0 0 680 16.50 21.1 5 5 560 1120 4 5
162 25 16.2 0 450 19.70 39.8 6 6 900 1800 4 6
80 20 8.0 0 370 22.26 71.2 3 3 170 340 2 3
85 25 8.5 0 480 27.74 7.9 3 3 260 520 2 3
55 10 5.5 0 660 25.92 41.3 1 1 30 60 0 1
55 10 5.5 0 665 27.27 22.2 1 1 30 60 0 1
55 10 5.5 0 670 27.79 17.3 1 1 30 60 0 1
Table 2 Forecasted demands, spinning reserves and spot market prices for 10-unit system. Hour
Demand (MW)
Reserve (MW)
Spot price ($/MW h)
Hour
Demand (MW)
Reserve (MW)
Spot price ($/MW h)
1 2 3 4 5 6 7 8 9 10 11 12
700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500
70 75 85 95 100 110 115 120 130 140 145 150
22.15 22.00 23.10 22.65 23.25 22.95 22.50 22.15 22.80 29.35 30.15 31.65
13 14 15 16 17 18 19 20 21 22 23 24
1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800
140 130 120 105 100 110 120 140 130 110 90 80
24.60 24.50 22.50 22.30 22.25 22.05 22.20 22.65 23.10 22.95 22.75 22.55
Table 3 Simulation results obtained for cooperative energy environment. No of units
10 20 40 60 80 100 150 200 300 400 500
GA
LR
LREA
Average cost ($)
Elapsed time (s)
Average cost ($)
Elapsed time (s)
Average cost ($)
Elapsed time (s)
565,825 1,126,249 2,251,981 3,383,631 4,504,989 5,587,538 8,405,341 11,175,072 16,762,600 22,350,152 27,937,680
823 1443 2411 3912 5181 8348 10,750 16,690 24,043 33,081 40,001
565,825 1,120,660 2,258,510 3,394,056 4,526,043 5,657,290 8,667,823 11,304,567 17,002,344 23,000,234 29,023,445
117 210 382 592 890 1312 1603 2102 3201 4322 5023
565,825 1,120,660 2,248,700 3,367,902 4,492,012 5,510,320 8,400,123 11,110,343 16,053,852 21,943,688 26,707,993
1232 1301 1352 1501 1981 2568 3023 4234 5983 6502 7034
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Table 4 Power setting of each unit of 10 unit system in cooperative energy environment. Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Power settings (MW) Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455 455
245 295 370 455 390 360 410 455 455 455 455 455 455 455 455 310 260 360 455 455 455 455 420 345
0 0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 0 0 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 0 0 0
0 0 25 40 25 25 25 30 85 162 162 162 162 85 30 25 25 25 30 162 85 145 25 0
0 0 0 0 0 0 0 0 20 33 73 80 33 20 0 0 0 0 0 33 20 20 0 0
0 0 0 0 0 0 0 0 25 25 25 25 25 25 0 0 0 0 0 25 25 25 0 0
0 0 0 0 0 0 0 0 0 10 10 43 10 0 0 0 0 0 0 10 0 0 0 0
0 0 0 0 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0
Table 5 Power setting of each unit of 10 unit system in competitive energy environment. Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Power settings (MW)/reserve settings (MW) Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0
245/70 295/75 395/60 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 455/0 415/40 455/0 455/0 455/0 455/0 455/0 445/10 345/80
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 130/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 162/0 162/0 162/0 162/0 130/32 160/2 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 68/12 80/0 80/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
For competitive environment
Minimize N X F i ðtÞ
Simulation studies and results
ð35Þ
i¼1
Subject to energy requirement, reserve requirement and generation unit limits. Typical Lambda-iteration method [1] is used to solve the economic dispatch problem. The iteration continuous until it meets the convergence criteria. Here the relative duality gap is used as the convergence criteria.
Relative Duality Gap ¼
Jq q
ð36Þ
Initially, some simulation studies [21,22] were carried out on test systems for benchmarking of the proposed algorithm. Then the feasibility of the proposed algorithm is demonstrated on power systems with much larger number of units. Cooperative environment The feasibility of the proposed algorithms is demonstrated with different size of power systems containing 10–500 units and the test results are compared with each other in terms of solution quality and convergence characteristic. Characteristics of units and load for the 10-units system are given in Tables 1 and 2.
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10 20 40 60 80 100 150 200 300 400 500
GA
LR
LREA
Average profit ($)
Elapsed time (s)
Average profit ($)
Elapsed time (s)
Average profit ($)
Elapsed time (s)
107,872 215,621 431,361 644,676 837,729 1,038,262 1,577,493 2,075,654 3,115,765 4,156,760 5,209,568
752 1323 2321 3821 5099 8235 11,243 16,345 23,654 31,456 42,497
107,872 215,743 431,223 642,122 812,499 997,924 1,528,782 2,000,234 3,109,456 4,098,234 5,198,374
111 198 368 571 851 1298 2046 3294 4594 5987 7943
107,875 215,747 431,489 647,235 862,979 1,078,718 1,678,332 2,164,749 3,278,398 4,387,573 5,457,975
1112 1230 1365 1498 1954 2537 3021 4364 5868 7365 9023
Spinning reserve of the system is assumed to be 10% of the load. The different number of units systems is created by repeating each unit of the original problem by respective factors. Thus, test problems with different numbers units are obtained. Hourly load and reserve amounts are also scaled by the same factors. After proper tuning the parameters of the proposed algorithm, simulation studies were carried out. The best results obtained by the algorithm are tabulated in Table 3. Further, Table 4 shows the power settings of each unit in case of 10 unit system when LREA is used. The proposed algorithm provides better numerical convergence for all the systems considered in the study. Currently, LR method is a widely accepted algorithm in real power system scheduling which is considered as a benchmark for comparing the results. It is shown in the paper that LREA provides better solutions. Competitive environment Similar to the cooperative environment, the feasibility of the proposed algorithms is demonstrated in competitive environment with different size of power systems containing 10–500 units. The same ten unit system is used in these studies also. Forecasted loads, spot prices, and reserve prices for the system are given in Table 2. In these studies, reserve prices are fixed at five times the spot prices, and probability that reserve is called and generated is kept at 0.05. Simulations were carried out and $113134.12 profit is achieved by the proposed algorithm. It is higher than that from [22]. The corresponding power settings of each unit are given in Table 5. Lagrangian relaxation and genetic algorithm are also used for generation scheduling of the same systems and used as benchmark algorithms. The best results obtained by the algorithm are tabulated in Table 6 for other systems containing different number of units. The proposed LREA algorithm provides better numerical convergence for all the systems considered in the simulation, and provides better solutions. LR for unit commitment problem provides a fast solution but it may suffer from numerical convergence and solution quality and Evolutionary Algorithm (EA) methods find the better solution but they need more computational time. In addition, LR decomposition procedure is dependent on the initial estimates of the Lagrangian multipliers and on the method used to update the multipliers. Currently the techniques used for estimating the Lagrangian multipliers rely on a sub-gradient algorithm or heuristics. Incorporating evolutionary algorithm into LR method to update the Lagrangian multipliers improves the performance of LR method in solving the unit commitment problem. As the result, it provides better numerical results obtained in a reasonable computational time.
Conclusion This paper proposes a hybrid algorithm to solve the short-term generation scheduling problem with operational constraints in a traditional and a restructured power systems using Lagrangian relaxation and evolutionary algorithm. In the restructured power system, spot prices and reserve prices in the market are important parameters while solving the profit based unit commitment problem. Using the proposed LREA approach, the generation companies can maximize their profit and schedule the generating units accordingly. The results achieved are quite encouraging and indicate the feasibility of the proposed technique to deal with large unit commitment problems in a deregulated environment. Furthermore, the proposed algorithm can find the most economical scheduling plan for traditional power systems according to the results. Using the proposed approach, the system can minimize the operating cost of generators substantially. According to the numerical results, LREA seems to be the best algorithm among the algorithms discussed in this paper for short-term generation scheduling problems in both cooperative and competitive energy environments.
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